齿轮系统的混沌控制及仿真
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摘要
基于混沌动力学的混沌控制研究是当前非线性动力学的前沿问题。本文对综合考虑时变刚度、间隙非线性的齿轮系统的混沌动力学行为及其混沌控制问题进行了系统、深入的研究，取得了一些成果和进展，论文主要内容如下：
1．针对单级齿轮系统，分别建立了具有分段线性特征的单自由度非线性动力学模型、综合考虑齿侧间隙和时变啮合刚度的三自由度非线性动力学模型，利用数值积分方法计算了系统的混沌响应，计算了系统的Lyapunov指数谱。
2．研究了增量谐波平衡法在具有分段线性特征的齿轮系统中的应用和对分段线性项的特殊处理过程，以及利用Floquet理论对周期解的稳定性和分岔类型的进行判定的方法。分析了激励频率，轴承阻尼这些参数对系统动力学特性的影响，绘制了系统随激励频率变化的响应曲线，观测到系统通向混沌的倍周期分岔和拟周期分岔路径，以及导致系统出现多重稳态周期解的鞍。结分岔。研究表明，IHB方法计算得到的周期解能很好地与采用数值积分方法计算得到的结果相吻合。算例展示了IHB方法在求解具有分段线性特征的三自由度齿轮系统周期解的有效性和准确性。
3．探讨了传统OGY方法的基本原理及其与现代控制理论的关系，证明了OGY控制方法其实质是一种特殊的极点配置技术；研究了如何将传统的OGY混沌控制方法拓展应用在齿轮系统这样的连续系统上，包括通过打靶法计算目标周期轨道位置，通过微分方程的变分形式计算目标周期轨道点处的Jacobi矩阵和敏感度向量；通过数值仿真成功地实现了单自由度齿轮系统多周期轨道的稳定化；给出了详尽的单自由度齿轮系统混沌控制的效果分析。
4．在对齿轮系统混沌吸引子内部不稳定周期解分析的基础上，提出一种多步混沌控制的方法。分析表明齿轮系统在一定参数区域具有典型的非双曲性质，表现在周期轨道点处的不稳定维数和稳定维数变异、Jacobi矩阵存在位于单位圆上的复共轭特征值。因此针对不稳定维数和稳定维数发生变异的两种情况，分别建立相应的混沌控制策略，通过连续的参数扰动将系统状态驱动到周期轨道的局部稳定流形上。多步控制方法本质上是经典OGY控制算法在高维非双曲动力系统中的推广，并解决了具有复共轭特征值的长周期轨道控制的难题。
5．利用延迟坐标技术建立了三自由度齿轮系统的混沌控制方案。通过对系统状态标量时间序列的分析，分别利用平均互信息函数方法和平均伪最近邻点方法获得了重构齿轮系统相空间的最优延迟时间和最小嵌入维数，建立了构建齿轮系统延迟坐标向量的完整方案。提出了在状态加参数系统中实现周期轨道稳定化的混沌控制算法，研究了从实测时间序列获得实现控制所需的各项参数的数据分析方法。数值仿真结果证实了混沌控制方法的有效性。
6．在齿轮动力学试验台上对齿轮系统的动态响应进行了测试，与理论计算结果进行了比较，证明了分析模型和计算机方法的有效性。
Chaos control research based on the chaotic dynamics is the frontier problem inmodem nonlinear dynamics. The chaotic dynamics and chaos control issue ofnonlinear geared system with time-varying mesh stiffness and backlash has beenextensively studied in this dissertation, the main research works are as follows:
1. For single-stage geared rotor-bearing system, a single-degree-of-freedomnonlinear dynamics model with piecewise linearity characteristic, and a3-degree-of-freedom nonlinear dynamics model involving gear backlash andtime-varying mesh stiffness are respectively developed. The chaotic responses of thetwo systems are calculated through numerical integration. The Lyapunov exponentspectrum of 3-d.o.f system is also calculated and plotted.
2. The application and the special treatment process for piecewise linear items ofincremental harmonic balance method(IHB) in solving periodic motions of gearedsystem are studied. The method to determine the stability of the periodic motion andits bifurcation types by Floquet theory is also investigated. The effects of theparameters such as exciting frequency and bearing damping on the system dynamicsare studied, the response diagrams for exciting frequency are plotted. Period-doublingbifurcation route to chaos, quasi-periodic bifurcation route to chaos and multiplesteady state periodic solutions caused by saddle-node bifurcation are observed in someparameter regions. It is shown that IHB method can be used to seek both the stable andunstable periodic solutions which compare very well with the results obtained usingnumerical integration. The example shows the effectiveness and accuracy of IHBmethod for computing the periodic solutions of 3-d.o.f. geared system with piecewiselinearity characteristic.
3. The principle of traditional OGY chaos control method and its relationship withmodem control theory are studied. It has been proved that OGY method is a specialpole placement technique in essence. The procedures to extend and adapt the OGYmethod into continuous system such as geared system are investigated, including theprocess of locating unstable period orbits(UPO) by shooting method, computing the Jacobian matrices and sensitivity vectors by variational form of differential equations.The stabilization of high period UPOs is achieved through numerical simulation. Theextensive effect analysis of chaos control of s.d.o.f geared system is presented.
4. A multi-step chaos control method is presented to realize the stabilization ofUPOs embedded in the chaotic attractor of 3.d.o.f. geared system. By means of theanalysis of the structure of system's UPOs. the existence of complex-conjugateeigenvalues on the unit circle along UPOs and unstable dimension or stable dimensionvariability of the UPOs indicate the nonhyperbolicity of the geared system. The controlmethod is set up for the situations of unstable dimension and stable dimensionvariability respectively to drive the system state to lie on the local stable manifold ofUPOs through continuous parametric perturbations. The method is essentially thegeneralization of classical OGY method in nonhyperbolic and high-dimensionalsystem. Numerical simulation verifies the effectiveness of the control method for theUPOs even with long period.
5. Assuming the prior knowledge of the geared system's dynamics is unknown, acomplete chaos control scheme is derived using delay coordinates technique. Theoptimal delay time and minimum embedding dimension for phase space reconstructionare acquired respectively by averaged mutual information method and averaged falsenearest neighbors method, the whole scheme for constructing delay coordinate vectorsof geared system is set up. Considering the dynamical dependence on the pastparameters, the chaos control method is constructed based on the combined dynamicsof state-plus-parameter system, the procedures to obtain the parameters forimplementing chaos control from experimental time series are investigated. Numericalsimulation shows the effectiveness of the method in practical applications.
6. The dynamic response of geared system is tested in a testing machine ofnonlinear spur gear pair. The vibration acceleration of the gear pair in differentparameter conditions is obtained in the testing process. The testing results are analyzedand compared with the theoretical results, which verifies the effectiveness of theanalyzing model and the calculating methods.
This work is supported by the National Natural Science Foundation of China (No.50075070).
1．针对单级齿轮系统，分别建立了具有分段线性特征的单自由度非线性动力学模型、综合考虑齿侧间隙和时变啮合刚度的三自由度非线性动力学模型，利用数值积分方法计算了系统的混沌响应，计算了系统的Lyapunov指数谱。
2．研究了增量谐波平衡法在具有分段线性特征的齿轮系统中的应用和对分段线性项的特殊处理过程，以及利用Floquet理论对周期解的稳定性和分岔类型的进行判定的方法。分析了激励频率，轴承阻尼这些参数对系统动力学特性的影响，绘制了系统随激励频率变化的响应曲线，观测到系统通向混沌的倍周期分岔和拟周期分岔路径，以及导致系统出现多重稳态周期解的鞍。结分岔。研究表明，IHB方法计算得到的周期解能很好地与采用数值积分方法计算得到的结果相吻合。算例展示了IHB方法在求解具有分段线性特征的三自由度齿轮系统周期解的有效性和准确性。
3．探讨了传统OGY方法的基本原理及其与现代控制理论的关系，证明了OGY控制方法其实质是一种特殊的极点配置技术；研究了如何将传统的OGY混沌控制方法拓展应用在齿轮系统这样的连续系统上，包括通过打靶法计算目标周期轨道位置，通过微分方程的变分形式计算目标周期轨道点处的Jacobi矩阵和敏感度向量；通过数值仿真成功地实现了单自由度齿轮系统多周期轨道的稳定化；给出了详尽的单自由度齿轮系统混沌控制的效果分析。
4．在对齿轮系统混沌吸引子内部不稳定周期解分析的基础上，提出一种多步混沌控制的方法。分析表明齿轮系统在一定参数区域具有典型的非双曲性质，表现在周期轨道点处的不稳定维数和稳定维数变异、Jacobi矩阵存在位于单位圆上的复共轭特征值。因此针对不稳定维数和稳定维数发生变异的两种情况，分别建立相应的混沌控制策略，通过连续的参数扰动将系统状态驱动到周期轨道的局部稳定流形上。多步控制方法本质上是经典OGY控制算法在高维非双曲动力系统中的推广，并解决了具有复共轭特征值的长周期轨道控制的难题。
5．利用延迟坐标技术建立了三自由度齿轮系统的混沌控制方案。通过对系统状态标量时间序列的分析，分别利用平均互信息函数方法和平均伪最近邻点方法获得了重构齿轮系统相空间的最优延迟时间和最小嵌入维数，建立了构建齿轮系统延迟坐标向量的完整方案。提出了在状态加参数系统中实现周期轨道稳定化的混沌控制算法，研究了从实测时间序列获得实现控制所需的各项参数的数据分析方法。数值仿真结果证实了混沌控制方法的有效性。
6．在齿轮动力学试验台上对齿轮系统的动态响应进行了测试，与理论计算结果进行了比较，证明了分析模型和计算机方法的有效性。
Chaos control research based on the chaotic dynamics is the frontier problem inmodem nonlinear dynamics. The chaotic dynamics and chaos control issue ofnonlinear geared system with time-varying mesh stiffness and backlash has beenextensively studied in this dissertation, the main research works are as follows:
1. For single-stage geared rotor-bearing system, a single-degree-of-freedomnonlinear dynamics model with piecewise linearity characteristic, and a3-degree-of-freedom nonlinear dynamics model involving gear backlash andtime-varying mesh stiffness are respectively developed. The chaotic responses of thetwo systems are calculated through numerical integration. The Lyapunov exponentspectrum of 3-d.o.f system is also calculated and plotted.
2. The application and the special treatment process for piecewise linear items ofincremental harmonic balance method(IHB) in solving periodic motions of gearedsystem are studied. The method to determine the stability of the periodic motion andits bifurcation types by Floquet theory is also investigated. The effects of theparameters such as exciting frequency and bearing damping on the system dynamicsare studied, the response diagrams for exciting frequency are plotted. Period-doublingbifurcation route to chaos, quasi-periodic bifurcation route to chaos and multiplesteady state periodic solutions caused by saddle-node bifurcation are observed in someparameter regions. It is shown that IHB method can be used to seek both the stable andunstable periodic solutions which compare very well with the results obtained usingnumerical integration. The example shows the effectiveness and accuracy of IHBmethod for computing the periodic solutions of 3-d.o.f. geared system with piecewiselinearity characteristic.
3. The principle of traditional OGY chaos control method and its relationship withmodem control theory are studied. It has been proved that OGY method is a specialpole placement technique in essence. The procedures to extend and adapt the OGYmethod into continuous system such as geared system are investigated, including theprocess of locating unstable period orbits(UPO) by shooting method, computing the Jacobian matrices and sensitivity vectors by variational form of differential equations.The stabilization of high period UPOs is achieved through numerical simulation. Theextensive effect analysis of chaos control of s.d.o.f geared system is presented.
4. A multi-step chaos control method is presented to realize the stabilization ofUPOs embedded in the chaotic attractor of 3.d.o.f. geared system. By means of theanalysis of the structure of system's UPOs. the existence of complex-conjugateeigenvalues on the unit circle along UPOs and unstable dimension or stable dimensionvariability of the UPOs indicate the nonhyperbolicity of the geared system. The controlmethod is set up for the situations of unstable dimension and stable dimensionvariability respectively to drive the system state to lie on the local stable manifold ofUPOs through continuous parametric perturbations. The method is essentially thegeneralization of classical OGY method in nonhyperbolic and high-dimensionalsystem. Numerical simulation verifies the effectiveness of the control method for theUPOs even with long period.
5. Assuming the prior knowledge of the geared system's dynamics is unknown, acomplete chaos control scheme is derived using delay coordinates technique. Theoptimal delay time and minimum embedding dimension for phase space reconstructionare acquired respectively by averaged mutual information method and averaged falsenearest neighbors method, the whole scheme for constructing delay coordinate vectorsof geared system is set up. Considering the dynamical dependence on the pastparameters, the chaos control method is constructed based on the combined dynamicsof state-plus-parameter system, the procedures to obtain the parameters forimplementing chaos control from experimental time series are investigated. Numericalsimulation shows the effectiveness of the method in practical applications.
6. The dynamic response of geared system is tested in a testing machine ofnonlinear spur gear pair. The vibration acceleration of the gear pair in differentparameter conditions is obtained in the testing process. The testing results are analyzedand compared with the theoretical results, which verifies the effectiveness of theanalyzing model and the calculating methods.
This work is supported by the National Natural Science Foundation of China (No.50075070).
引文
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